User’s Manual v 1.7

Remcom RLD v1.7

Table of Contents

1 – Overview

2 – Installation

3 – User Interface

4 – Program Usage – Design Process

5 – XFdtd Interface

6 – Design Example

7 – Theory and Background

8 – References

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1. Rotman Lens Designer – Overview

Remcom Rotman Lens Designer RLD is a computer-aided design program for the design, synthesis, and analysis of Rotman Lenses and their variants. It is based on Geometrical Optics combined with the classical Rotman Lens design equations as in [1,2]. It is intended for rapid development and analysis of Rotman Lenses given several physical and electrical input parameters. RLD generates the proper lens contours, transmission line geometry, absorptive port (dummy port) geometry, provides an approximate analysis of performance, and generates geometry files for import into Remcom XFdtd for further analysis. A CAD file export option may be used to export the lens design to other software, including converters for the creation of Gerber photo plotter files

RLD Lens Input Parameters

Input parameters include:

• Type of Lens – microstrip or stripline• Element spacing on the antenna• Number of Elements (up to 128)• Number of beams (up to 128).• Maximum scan angle.• Off-Axis Focii Angle as a fraction of Max Scan Angle

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• Bandwidth and Center Frequency• Electrical characteristics of lens material - dielectric constant, loss tangent,

conductivity and thickness of dielectric material.• Electrical characteristics of absorber material • Physical characteristics of lens – Focal Length/Diameter Ratio• Aperture distribution• VSWR of ports

RLD Output Capabilities

The Rotman Lens Designer is capable of providing the following results:

• Calculation of the lens contour and of the transmission line geometry• Phase errors• Element locations along the lens contour• Insertion loss (per beam port)• Amplitude and phase distributions of the array ports versus frequency.• Suggested Focal Length• Number of dummy elements• Location of loads on dummy elements• Location of absorber if necessary • Creation of a CAD output file• Dimensions, volume• Suggested XFdtd cell sizes

Since the intended usage for RLD is rapid synthesis of microwave lenses, various approximations and assumptions are made to allow for “real-time” design and tuning of the lens structure.

The analysis is based on geometry and geometrical optics and therefore assumes:

• Radiative leaks are not accounted for• Transmission line and material dispersion is negligible• Dummy load effectiveness is assumed ideal

Effects which are accounted for include:

• Direct and Single Reflection propagation between ports including sinc(u) radiation patterns for port tapers.

• Calculations of ray amplitude between ports include dielectric and conductor losses.• Individual port VSWR is approximated as that of the port to transmission line

transition.• Individual port losses are accounted for using analysis for the type of transmission

lines used.

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Rotman Lens Layout and Nomenclature

Remcom RLD uses standard layout nomenclature and port numbering as is done in the literature [1,2,6,7] The following diagram points out the primary components and their various names and possible variations. Throughout this manual and in the RLD program these definitions are adhered to. RLD synthesizes lenses by generating the beam and array contours, allows the user to form the dummy or absorber contours, and creates the proper transmission lines to form a functioning Rotman-type lens. Starting with Version 1.3, the software has the capability of including transmission lines for the beam and dummy ports. If the transmission lines are not included, the tapered port contours do provide a connection point for transmission lines at the user defined system impedance.

Port Numbering

Another important item to include in the description of the Rotman lens layout is the port numbering scheme.

The ports along the Rotman lens contour are divided into two groups, active and passive. All ports are ordered in a single continuous list to maintain port index uniqueness. Beam and Array ports are Active ports while Dummy ports are inactive. The port numbering starts with Beam port 1 on the lower left side of the lens and moves clockwise around the Beam contour counting active ports. If dummy ports are encountered along the Beam contour they are not counted. Continuing clockwise after the last Beam port is counted, the first active port on the Array contour is counted continuing down the Array contour to the last active element on the lower right of the lens. Then, continuing clockwise the counting of inactive dummy ports begins counting any dummy ports within the Beam contour, continuing clockwise along the upper dummy contour, down the Array contour counting any dummy ports within the Array

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contour and finally the lower dummy contour from right to left. This is shown pictorially below:

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2. Rotman Lens Designer – InstallationThe RLD program requires the capabilities of a workstation or powerful PC. At this time it is available for the Linux and Windows (2000, XP, Vista, Windows7) operating systems.

CD Contents

The contents of the RLD CD include the RLD Software installation packages and the RLD Manual.

System Requirements

Pentium III 800 MHz or better 1024x768 resolution (1280x1024 or higher preferred)

Minimum 256 colors OpenGL compatible graphics

At least 256 MB of memory

Installing the Software - Windows

Run RLD_1.7_setup.exe and follow the instructions. This will create a Start Menu item for RLD under “Remcom”.

Installing the Software - Linux

Unpack the file RLD_1.7.tgz. A “remcom” directory will be created with an “RLD_1.7” directory inside of it. If you have other Remcom products installed, you should unpack the file in the directory above the already existing “remcom” directory. For example, if you have

/opt/remcom/XFdtd_7.1.2.3

you should unpack RLD using commands similar to:

cd /opttar -zxvf /mnt/cdrom/Linux/RLD_1.7.tgz

Run RLD by executing the command (following the example above):

/opt/remcom/RLD_1.7/Linux-i686RHEL5/rld

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Acquiring a License

To acquire a license for RLD, start RLD. If you do not already have a license, RLD will display the dialog:

Clicking “OK” will bring up the License Dialog:

If you have not already received a license file from Remcom for RLD, send the Host ID listed in the dialog or a screenshot of the dialog to license@remcom.com. When you have a valid license file, either place it in the default directory shown in the dialog or specify the file directly by clicking on “Browse...” and navigating to the license file. Click “Apply” when finished. If the license is found, the “OK” button will become enabled. The application will start when the “OK” button is clicked. This window will appear each time you start RLD unless “Do not show this window at startup” is checked. This behavior can be overridden in the preferences window in RLD.

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The licensing dialog can be reopened at any time by selecting “Licensing...” from the “Help” menu on the main menu bar of the application.

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3. Rotman Lens Designer – User InterfaceRemcom RLD has an intuitive but comprehensive user interface for the design and analysis of Rotman Lenses. The various parameter entry fields and parameter controls have been arranged in a tab system to help organize the work flow of microwave lens design.

RLD is available for both Windows and Linux so most of the generic features will seem familiar to even casual pc users.

File Bar Features

Along the top of the RLD main window are the File bar menus for File,View,Tools, and Help

Clicking the File menu will open a menu containing:

File->Open – Opens an existing Rotman lens design, user will be prompted for lensFile->New Lens - Opens a new Rotman lens with default parameter settingsFile->Save - Saves the currently active Rotman lens design, user will be prompted for filename if it has not been saved beforeFile->Save As - Saves the currently active Rotman lens design, user will be prompted for a new filenameFile->Export – Has four options:

Exports the currently active lens as an SAT format CAD fileExports the currently active lens asr a geometry file in 2ds format for importation into

Remcom XFdtd® Exports the S-parameter data of the currently active lens. User will be prompted for

Beam port number. Insertion Loss may also be exported and again user will be prompted for Beam port

number.

File->Preferences – Open a dialog box where various user preferences can be modified and saved.

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Window Layout – Remember Layout on program Exit checkbox will control RLD to save the window layout on program exit and will return to this state when the program is re-opened.Remember Layout Now saves the current window layout.

Forget Layout Now clears any history of Layout.

Startup- Show License Selection Window causes the license choice window to appear at next startup.

Set Default causes the preferences to return to the default as-shipped settings.

File->Close – Closes the current active lens, will prompt user if Save is necessaryFile->Quit – Closes all lenses, Exits RLD, will prompt user if Save is necessary

Clicking the View menu will open a menu containing:

View ->Messages Checkbox – Toggles the view of the Message box. This box is used to display warnings and other info to the user during lens design. A warning indicator will appear in the lower left side of the lens window if messages are available regardless of the state of this toggle.

View->Port Info Checkbox – Toggles the view of Port info as the mouse cursor is moved over a port. All ports will display the type and port number. Beam ports will display the excitation currently applied to them.

Clicking the Tools menu will open a menu containing:

Tools-> Plot-> Array Factor - Plots the array factor for the current lens and defined excitation.

Tools-> Plot-> Beam Ports->

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Beam To Array Phase Error - Open a plot for phase error over the array ports from a given beam port or ports.

Beam to Array Coupling Magnitude - Open a plot displaying the array ports amplitude distribution from a given beam or set of beams.

Beam to Array Coupling Phase - Open a plot displaying the array ports phase distribution.

Beam to Sidewall Coupling Magnitude - Open a plot displaying the coupling of energy from the beam ports into the sidewalls.

Beam to Array Spillover Coupling Magnitude - Open a plot displaying the coupling of energy from the beam ports to array ports due to sidewall reflection.

Tools-> Plot-> Array Ports ->

Array to Beam Coupling Magnitude - Open a plot displaying the beam ports amplitude distribution from a given array port or set of array ports.

Array to Beam Coupling Phase - Open a plot displaying the array ports phase distribution.

Beam to Sidewall Coupling Magnitude - Open a plot displaying the coupling of energy from the array ports into the sidewalls.

Beam to Array Spillover Coupling Magnitude - Open a plot displaying the coupling of energy from the array ports to beam ports due to sidewall reflection.

Tools-> Plot-> Post Processing -> Insertion Loss - Opens a plot of Lens insertion loss for a given set of beams or single beam.

Tools-> Plot-> Post Processing -> S-Parameters - Opens a plot of Lens scattering parameters for a given set of beams or single beam.

These plots are described in full detail later in section 3.

File->Help

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By clicking Help and clicking What’s This ? (or Shift F1) as shown above, the mouse cursor will turn into a question mark and can be used to display a help dialog for any item in the RLD interface and the parameters.

Double-clicking the RLD icon will open the RLD graphical user interface (GUI) shown below. To begin using RLD, the user must either open an existing lens file or create a new lens. To do this click on the File Open Icon or New file Icon shown below. Alternatively the user can click File->Open, or File->New Lens from the menu items along the toolbar or Click CTRL+N for a new lens or CTRL+O to open an existing Lens.

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Moving the mouse cursor over any of the buttons in RLD will reveal a tool-tip explanation of the button as shown below.

Once the New Lens Icon is clicked RLD will load a default Rotman Lens. A set of “Tabs” will appear which contain parameters, controls, and information about the loaded lens. Each lens that is loaded will add a new tab to the Upper tab list and each will contain its accompanying control tabs, Physical Properties, Electrical Properties, TxLine Properties, Description. Clicking any tab brings it to the front and makes this tab active as shown below. The right or left arrow can be used to slide over to hidden tabs.

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The following are descriptions of each of the sub-tabs associated with each lens.

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Physical Properties Tab

System Z (Ohms) - Chooses the impedance which is to be used to connect the inputs and outputs of the Rotman lens.

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Lens Type - Chooses the medium to design the lens in. Primarily affects the connecting transmission line structure and behavior.

Focal Contour Shape – Choose the shape of the focal or “beam port” contour. Classical Rotman Lens designs use circular focal contours. For large lenses, phase error can be reduced by using an Elliptical contour [2].

Focal Length– The distance between the center of the Array contour and the center of the focal contour (G). This value may be entered in dielectric wavelengths or meters. If the Auto box is checked, RLD will automatically select a nominal value for the focal length to minimize phase error based on the current lens. Further changes to the lens geometry or specifications will update this value. This value can be modified if needed by un-checking the Auto box.

Focal Ratio (g) – The ratio of on axis focal length to off axis focal length [1] or G/F1. It is used to control the shape of the Focal arc [1]. Valid range is ~(0.9,1.2)

-X

α

ε rε 0

F1

Beam or Focal Arc

Inner or Array Contour

G

Y

Focal Points

Substrate

Loss Tangent – The loss tangent tanδ of the lens dielectric. Used in calculation of port coupling and Array Factor.

Dielectric Constant – The dielectric constant ε of the lens substrate. This material supports both the parallel plate region and the transmission lines. The upper limit of the dielectric constant is set at 50 as this is the limit of validity for the equations in the software.

Thickness (mm) – The thickness of the dielectric substrate. It is the separation between the lens contour and is groundplane(s) in the z-dimension.

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Density (g/mm^3) – The density of the dielectric substrate. It is used in the calculation of the lens weight.

Metallization

Conductivity – The conductivity σ of the lens metallic surfaces. Used in calculation of insertion loss.

Thickness (mm) – The thickness of the lens metallic surfaces. This dimension is used for both the parallel plate region and the transmission lines.

Absorber

Dielectric Constant – The dielectric constant ε of the lens absorber if used.

Conductivity – The absorber conductivity.

Lens Info

This area provides data regarding the current active lens and its current state. Changes to the lens will automatically update this information. This information includes:

• Loaded From - Reports which file was loaded to produce this lens• Focal Length• Ports (beam, array, dummy) – total reports the statistics on the number of each type

of port• Approximate VSWR for each type of port. This is estimated form the port flare

angle.• Dimensions of the lens including the linear array• Lens Volume• Lens Weight• Approx. FDTD cell size – reports a suggested maximum cell size for use in XFdtd

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Electrical Properties TabThis tab primarily contains parameters and information regarding Electrical performance and antenna parameters.

Center Frequency (MHz) - The design frequency for which the lens geometry is scaled. Enter in MHz.

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Bandwidth (MHz) – The operating range of frequency. Data versus frequency will be plotted over this range.

Element Spacing - The spacing of the linear array elements along the outer contour. The spacing is in free-space wavelengths or meters.

Beam Port Excitation - Allows the user to control how the beam port is excited for Array factor plotting. Uniform excites each port in the focal contour with a 1+j0 volt source. User defined allows the user to individually control each port or import the excitation information from a file (using the “Edit” button).

Once a distribution is set up, the user may wish to save it for use with other lenses. For Example, suppose the user wishes to have 7 beams with a tapered excitation ( -3dB,-6dB, -9dB). Click the Remove Port button until 7 ports remain, then Edit the Voltage magnitudes like the following:

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Click the Export button and give the distribution file a name,

Then finally click OK to accept this Beam port distribution.

Later, opening the User defined distribution and clicking Import will allow the user to open this saved distribution. The number of Beam ports will automatically adjust to the number

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specified in the distribution file. (*.rdist). The format for this file is simple, each line denotes a separate port number and contains magnitude, phase pairs (V Deg).

As an example, the following is the resulting taper7ports.rdist created above:

0.354 00.501 00.707 01 00.707 00.501 00.354 0

Aperture Distribution - This allows the user to select the desired aperture distribution on the array contour. Choices of Manual, Uniform, 3dB, 6dB, 9dB are provided. Manual allows the user to manually adjust the aperture taper by adjusting the beam port width (all beam ports the same), Uniform will automatically adjust the beam port width to create a uniform energy distribution along the array contour from each beam port. Likewise the 3, 6, 9dB tapers will automatically adjust the beam port widths (non-uniformly if required) to produce the desired aperture taper. If the geometry of the lens cannot support the required taper, a warning message will appear suggesting to the user, the proper adjustment or adjustments to allow the aperture taper.

Max Scan Angle (degrees), Alpha Ratio – Max Scan angle is the maximum angle the lens will be used to scan over and therefore the angle between the lens axis and the outermost beam port. The off-axis focal points at +/- alpha are placed at the fraction set by Alpha Ratio of the max scan angle. The off-axis focal points are at the off axis beam angle for which phase error is zero [1].

Beam Contour Controls

Number of Beams - Sets the number of beam the user wishes to control. Adds or removes focal contour ports and focal contour “dummy” ports based on the scan angle.The valid range is between 2 and 128 beams.

Max Port Size (wavelengths) - Sets the maximum size ( in guide wavelengths) a beam port can grow before it automatically reduces to allow the insertion of a dummy port. Adds or removes focal contour “dummy” ports based on the scan angle and number of beams.

Flare Angle (Degrees) – Allows adjustment of the flare angle of the beam ports.

Enable Port Pointing - Toggle to control whether the beam port orientation is normal to the Beam contour or oriented along the line from the beam port to the center of the Array Contour. See figure below.

Array Contour Controls

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Number of Elements - Sets the number of array elements the user wishes to control. Adds or removes array contour ports and array contour “dummy” ports based on the element spacing and focal ratio. The valid range is between 2 and 128 elements.

Max Port Size (wavelengths) - Sets the maximum size ( in guide wavelengths) an array port can grow before it automatically reduces to allow the insertion of a dummy port. Adds or removes array contour “dummy” ports based on the scan angle and number of elements. This is generally controlled by the lens equations.

Flare Angle (Degrees) – Allows adjustment of the flare angle of the array ports.

Enable Port Pointing - Toggle to control whether the Array port orientation is normal to the Array contour or oriented along the line from the array port to the center of the Beam Contour. See figure below.

Sidewalls

Absorber Sidewalls – Sets the sidewalls to absorber. The currently entered absorber parameters (on the Physical Properties Tab) will be used.

Contour Curvature - Controls the rate or curvature for the upper and lower contours between the focal and array contours.

Max Port Size (wavelengths) - Sets the maximum size ( in guide wavelengths) a dummy port can grow before it automatically reduces to allow the insertion of another dummy port. Adds or removes upper and lower contour “dummy” ports based on the curvature.

Flare Angle (Degrees) – Allows adjustment of the flare angle of the dummy ports, if used.

Absorber Width Factor – Allows adjustment of the absorber width, if absorber is selected.

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TxLine Properties TabThis tab contains parameters and information regarding the transmission lines attached to all the ports.

The first check box at the top of the tab labeled “Calculation Lines Geometry” allows the user to turn off the computation of the transmission lines until the lens has satisfied other performance criteria. This is due to the small computational overhead of re-computing the lines each time the lens is modified.

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The first type of lines considered are the Array transmission lines attached to the Array ports at the output side of the lens. These lines may be entered in one of two way, controlled by the Line Type entry. This menu allows the user to specify if the lines are to be connected to a radial array of connectors or directly to a linear array of antenna elements. For the first case, Straight lines connected in a radial array, the appropriate length of line is added for proper phasing and the user is allowed to add a fixed length of line (W0) to accommodate physical constraints. The assumption here is that the Rotman lens will be connected to the antenna system through equal length transmission lines, coaxial cables or some other type. For the second type of line, Routed, used for direct connection to a linear array, the connecting transmission lines from each array port to each array element are created and routed to provide the proper phase length for the medium in use, maintain smooth curvature to minimize reflections from discontinuities, and maintain proper distance between lines to minimize crosstalk.

The generation of these lines is semi-automatic with user assistance. RLD will create a double spline to provide the smooth path and provide the proper length. The Spread Factor adjusts the separation width between the lines. The position where the splines meet is adjusted by using the Intermediate Position Factor. Setting this to 1 will create a single spline curve and is generally all that is required. For tightly spaced lines with large numbers of elements, moving the Intermediate position midway between the array and the array port will help position the lines properly. For situations where there still is not enough space to facilitate proper lines, the Length Factor control can be used to add fixed amounts of line (a percentage of the maximum line length) to each of the lines to allow for proper routing. The final control Terminating Position Factor allows the array position with respect to the lens to be adjusted. This is sometimes required to allow the routing algorithm to work and meet all of the above requirements.

The Beam transmission lines have three control options. The Terminal Spacing control sets a separation distance in millimeters between the input port connectors. These connection points represent the locations for connectors on a circuit board. The Terminal Distance Factor adjusts the length of the transmission lines. The Terminal Line Curvature adjusts the smoothness of the line as it transitions from the port to the connection point. The three controls should be used in conjunction to add transmission lines to the beam ports. The transmission lines serve no function in terms of the RLD calculations, but they are included for use when exporting the geometry either to XFdtd or to a Gerber format.

The Dummy transmission lines have the same controls as the Beam transition lines discussed above. A fourth control, the Terminal Position Shift Factor may be used to shift the dummy port connectors as a unit.

Data Plotting and Saving

RLD has several types of data plots available to aid in the analysis and tuning of a Rotman lens. For each of the data plots available there are simple controls to zoom in and out or to window a particular region of data, these controls are shown below.

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Clicking will cause the mouse cursor to switch to zoom mode. Clicking and dragging on the data plot will zoom to the region selected. Simply click and hold, drag the dotted box out to the region of interest and release the mouse button. If this operation is repeated, all levels

of zoom are remembered and clicking will cause the cursor to enter un-zoom mode and

will un-zoom one level per click in the data plot until full view is reached. Clicking will return to the top level (un-zoomed) immediately without saving the zoom levels. This is shown in the Array factor plot that follows.

Each plot included in the RLD program may be saved into a text file by merely right clicking on the plot entry and selecting Save from the pop-up menu. Additionally plots may be removed from display and the line color may be changed with this menu. More details are given later in the section on Exporting Data from Plots.

The plots are available by clicking:

Tools-> Plot-> Array Factor - Plots the array factor for the current lens and defined excitation.

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Tools-> Plot-> Beam Ports->

Beam To Array Phase Error - Open a plot for phase error over the array ports from a given beam port or ports.

Beam to Array Coupling Magnitude - Open a plot displaying the array ports amplitude distribution from a given beam or set of beams.

Beam to Array Coupling Phase - Open a plot displaying the array ports phase distribution.

Beam to Sidewall Coupling Magnitude - Open a plot displaying the coupling of energy from the beam ports into the sidewalls.

Beam to Array Spillover Coupling Magnitude - Open a plot displaying the coupling of energy from the beam ports to array ports due to sidewall reflection.

Tools-> Plot-> Array Ports ->

Array to Beam Coupling Magnitude - Open a plot displaying the beam ports amplitude distribution from a given array port or set of array ports.

Array to Beam Coupling Phase - Open a plot displaying the array ports phase distribution.

Beam to Sidewall Coupling Magnitude - Open a plot displaying the coupling of energy from the array ports into the sidewalls.

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Beam to Array Spillover Coupling Magnitude - Open a plot displaying the coupling of energy from the array ports to beam ports due to sidewall reflection.

Tools-> Plot-> Post Processing -> Insertion Loss - Opens a plot of Lens insertion loss for a given set of beams or single beam. The data must first be exported.

Tools-> Plot-> Post Processing -> S-Parameters - Opens a plot of Lens scattering parameters for a given set of beams or single beam. The data must first be exported.

Exporting S-parameters

Once a lens has been adjusted and meets specifications, the user may want to export the lens s-parameters for post processing, analysis and comparison. By clicking File-> Export-> S-Parameters a dialog box to choose which parameters are to be saved as shown below.

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The location the s-parameter files will be placed is entered in to Output Directory which

can be edited using . It is generally a good idea to create a subdirectory as many files will be placed here.

The format for the name of the s-parameter data file is as follows:

BaseFileName.S_BeamPortNumber_ArrayPortNumber.sp

Where BaseFileName is derived from the lens base file name or can be edited as needed. BeamPortNumber is the Beam port to be excited, and ArrayPortNumber is the index of the array port.

By clicking the beam port for which s-parameters are desired and clicking the -> the dataset will be chosen and exported when export is clicked. Any or all of the beams on the left may be selected or deselected by clicking <-. Clicking an item and then holding the shift key (SHFT) or control (CTRL) keys and then clicking again will have standard windows behavior i.e. multiple select or single item picking.

Next the user must choose the Min,Max frequencies desired, and the number of frequencies. This is important for matching the input data requirements for post-processing software.

Once Export is clicked data files will be placed in the directory chosen.

The format for the s-parameter files is each file is for an Sij over the specified frequency range at the number of samples specified. The # symbol can be used for a comment line. For example, the data in a file named

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TestLens.S_2_7.sp

has the format:

#freq. Hz Re{s(2,7)} Im{s(2,7)} mag{ s(2,7)} (dB) phase{ s(2,7)} (deg)

3.44000e+09 -0.130003 -0.100266 -15.6938 -127.6423.45939e+09 -0.141038 -0.0847575 -15.6742 -121.0043.47879e+09 -0.150224 -0.0680383 -15.6549 -114.3663.49818e+09 -0.15743 -0.050329 -15.6357 -107.7293.51758e+09 -0.162548 -0.0318642 -15.6166 -101.0913.53697e+09 -0.165502 -0.0128897 -15.5976 -94.45333.55636e+09 -0.166242 0.00634093 -15.5789 -87.81563.57576e+09 -0.164749 0.0255693 -15.5602 -81.1783.59515e+09 -0.161032 0.0445361 -15.5417 -74.54033.61455e+09 -0.155132 0.0629845 -15.5233 -67.90263.63394e+09 -0.147121 0.0806634 -15.505 -61.2649...

Exporting Insertion Loss

The procedure for exporting the insertion loss is identical to that for the S-parameters discussed above. The insertion loss files have a naming convention of LENS_NAME.BeamPort_NUM.il where LENS_NAME is the name of the lens as saved and NUM is the beam port number selected. The insertion loss files have two columns, the first is the frequency in Hertz and the second is the insertion loss in dB.

Exporting Data from Plots

RLD has the ability to export the data in a curve for use in comparisons to other lenses or further analysis. With a plot open simply right-click on the legend entry for the curve to be saved and click Save. The user will be prompted for a name and location if the plot belongs to the current lens. Curves which were imported from other data sources are not owned by the lens which created the plot, so cannot be exported. At the bottom of the file save window there is an entry for Legend text. It is good practice to be explicit since this field will be used when importing into another graph.

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Importing Data

RLD has the ability to import data in a simple convenient text format for use in comparison

of results with measured or other data. In any open graph click the file open button and a browse window will appear to select the file to be imported. The data needs to be in a file with extension *.flat or *.rplot. This data can be other datasets from RLD (*.rplot) or user supplied data such as antenna range data, network analyzer data, or data from another computation. The only requirement is the data be in following simple form:

#This is to test a comment#This is another comment and the next line is whitespace

# This is the legend textTestPlot #It can have comments here too

# This is the plot typeunknown#Values can be# ArrayFactor, BTAPhaseError, BTACouplingMagnitude, BTACouplingPhase, #or Unknown Unknown is a valid type# note that only unknown can be loaded into any type of plot

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# The Data Values themselves as x,y pairs 0 01 1.22 2.553 3.014 4.025 5.056 3.017 2.098 2.09 1.2

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4. Rotman Lens Designer – Program Use

Typically the design of a Microwave Lens using RLD will start with a set of design requirements including frequency of operation, Number of Beams, Scan Angle, Array dimensions, etc. Along with these requirements, several physical and electrical constraints will be imposed to deal with system interfaces, physical size and shape limitations, available materials, etc. RLD allows for fast and easy entry of these parameters in the two tabs, Physical Properties, Electrical Properties mentioned above.

The typical procedure calls for determining the proper linear array dimensions to produce the beamwidth required, prevent grating lobes, and meet physical space requirements. RLD will use the center frequency of operation to calculate the dimensions entered in terms of wavelength, then the user must select a focal length, typically near but less than the array dimension, then iteratively adjust g (focal ratio), scan angle, element spacing to produce a lens with minimal phase errors, acceptable array factor and beam port arc of comparable size to the inner array contour. As adjustments are made to any of these parameters, RLD continuously updates the lens geometry and any performance plots that are open.

At any instance this may or may not be a valid lens geometry or may not yield the desired performance. Iterative adjustment and “tuning” of the slide controls and edit boxes is generally necessary to fine tune a particular lens design. This is depicted below in the “Rapid Tuning and Optimization Loop”.

Once the user has determined the geometry and performance meet or approximate the requirements, the lens geometry can be exported to CAD format for fabrication, or exported to XFdtd for final Full-Wave analysis and further CAD manipulation. If the user determines changes need to be made after the XFdtd analysis he or she has the option to make the changes directly in XFdtd or to return to RLD to make the necessary changes.

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5. Rotman Lens Designer – XFdtd Interface

To use RLD in conjunction with XFdtd it is assumed the user has a working knowledge of XFdtd. Please contact Remcom (License@remcom.com) for support in this area.

An XFdtd script has been written to load a 2ds file created by RLD into XFdtd. To use this script, start XFdtd and import the script file “ImportRLDGeometry.xmacro”. This is shown below.

After importing the script, simply execute it by right-clicking on the script name and selecting Execute from the menu. The script will begin and ask for the name of the 2ds file previously exported from RLD. Browse to select this file and then press Open as shown in the following image.

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Remcom RLD v1.7

After selecting the file, the script will begin adding geometry features to the project. The features will be drawn in the main geometry window and each piece will be added as a part in the Parts List at the left of the screen. Once all parts are added, the script will add the ports to the geometry at the ends of all the transmission lines. To do this, it will need to read the 2ds file again, so the same file selection window used previously will appear again. Simply select the same exported 2ds file and press Open as before. Another window will appear that asks for the number of beam, array, and dummy ports so they will be categorized correctly. Refer to the RLD geometry or the geometry on the XFdtd screen to enter the proper numbers. This window is shown in the following image.

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Remcom RLD v1.7

Please note that the dielectric substrate is added as a part to the geometry but the material parameters for the dielectric are not. It is the user’s responsibility to edit the substrate material to enter the electrical conductivity or loss tangent and dielectric permittivity. Also note that a ground plane is not added to the geometry. In typical use a perfectly conducting boundary condition may be used at the lower Z boundary in the geometry to function as the ground plane. If for some reason that is not desired, then the ground plane will need to be added as a separate geometry object. Also, the lens material is selected as perfectly conducting. If a finitely conducting material, such as copper, is desired the material may be added and assigned to the lens.

At this point the geometry should be entered exactly as it appeared in RLD and the ports defined. The structure is now ready for full-wave simulation or export to a supported CAD format file.

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6. Rotman Lens Designer – Design ExampleThis section goes through a specific design example, making use of all of the features of RLD. The example was taken from reference [32]. The requirements for the Rotman lens in this paper were:

• Microstrip configuration• Frequency 16 GHz • Bandwidth 4 GHz• Material RT-Duroid rε =2.33, 20 mil thickness• Less than 15 λ or 30x35 cm square area• Scan Angle 20 deg. • Number of Beams 7• Number of Elements 16

To begin open the RLD program by double clicking the icon or a shortcut. This will open the main window where the user can click New Lens and begin to enter the physical parameters. First, it is good practice to give the lens a name so the various output parameters which use the lens name will appear correct. To do this, click the Description Tab and enter a name such as is done below:

Clicking return will accept the value. Other information about the lens may be entered in the Lens description below such as the project engineer or special requirements.

Next, click the Physical Properties Tab. Note the asterisk* on the Tab name. This indicates the lens has changed and needs to be saved. Clicking the save button will cause the asterisk to disappear, and the lens has been saved.

After entering the system impedance (nominally 50 ohms), the lens configuration (Microstrip), The Focal Contour Shape may be left as Circular for now, use of this control is described in section 3 and also in [2]. Continuing, click the Auto check box if it is not checked, or enter a reasonable starting point for the Lens width (0.15 meters) since the requirement was for a total length ~35cm and this includes the transmission lines. A starting value of ~ 1.1 is used for the focal ratio g. The Rogers specification for RT5870 gives a Loss

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Remcom RLD v1.7

Tangent of 0.0005 with dielectric constant 2.33. The specified Thickness of 0.508 mm is also entered in the Substrate specification box. The groundplane and top surface metal (copper) conductivity and metallization thickness are entered in the Metallization box

At this point the user is ready to enter the Electrical parameters. To do this, click the Electrical parameters tab. In this tab, the user enters parameters involving the frequency of operation, the array, the beams, the ports and transmission lines. Begin by entering the center frequency and bandwidth of 16000 MHz and 4000 MHz. A nominal element spacing of 0.5 wavelengths, assume a uniform beam port excitation for now, then select the desired aperture distribution or select manual. This example uses a uniform aperture distribution. Next set the Max Scan Angle to 20 degrees and the Alpha ratio to approximately 0.8.

Next, slide the control or type in the edit box under Beam Contour to create 7 beams and thus 7 beam ports. If a particular aperture distribution was chosen, the beam port widths will be automatically adjusted (non-uniformly) to produce the desired taper along the array contour and thus the maximum port size control will be disabled. If Manual was chosen for Aperture Distribution, the Max Port Size control will be active and can be used to create

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Remcom RLD v1.7

particular aperture distributions or other lens characteristics. This setting controls all beam ports uniformly. Adjusting the maximum port size also controls how the focal arc is filled in with ports geometrically. Smaller ports may require dummy ports to be inserted between Beam ports to cover the arc.

Next, set the Flare Angle for the beam ports to a nominal level ( ~11 Deg.) such that the estimated VSWR is acceptable as well as the transmission line taper length.

The checkbox for Enable Port pointing will be used later for optimally pointing the beam ports at the array contour center. Checking this box will re-aim the beam ports from their arc-normal position to pointed at the array contour center point.

Similarly, set the number of Array ports to the specified 16. The RLD lens solver will automatically set the array port width but the user may find it useful to make them smaller and insert dummy port along the inner Array contour. The control cannot create array port larger than the Rotman lens was calculated for due to the geometric restriction on the phase center positions along the array contour calculated by the Rotman lens equations.

Finally, set the Sidewall parameters, by selecting whether absorber or dummy ports are used, the Sidewall contour curvature, the Max Port Size and Flare Angle, for dummy ports if used, and the Absorber Width Factor if absorber is used. Regarding the use of absorber, a couple of points are important. The number of absorber sections used in the sidewall reflection analysis can be controlled by the Max Port Size control by un-selecting absorber, adjusting the Max Port Size to produce the number of segments desired, then re-select absorber. Generally, if absorber is used, it is best to set this number as high as possible for best accuracy. Secondly, the absorber width factor is geometric only and is used to control the dimensions of the absorber polygon that will be exported to XFdtd.

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Remcom RLD v1.7

RLD has a set of controls to produce various transmission line configurations connecting the array ports to the array elements. To access this click the TxLines tab.During the set up and tuning phase of the design process the user has the option to view the lens transmission lines or not, controlled by the Calculate Line Geometry checkbox. Generally this can be turned on, with a slight reduction in response time for lenses with large numbers of ports. The next control is Line Type. This controls whether the transmission lines will be Routed to a linear array or project Straight out radially from the array contour. The assumption being the radial transmission lines will be connected to the array through a set of equal length coaxial cables or other equal length transmission lines.

If Routed is selected, the transmission lines will form to connect the array contour (inner lens contour) to the linear array (outer lens contour) for each corresponding element.

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The controls in the Transmission Lines tab are used to adjust the geometry of the transmission line routing to ensure no overlapping, proper spacing between lines, proper curvature and maintaining overall physical length requirement. A few minutes of using the controls will help familiarize the user with the function provided by each. The transmission lines will be exported in exactly the configuration shown on the screen.The line lengths are automatically adjusted to provide the exact amount of required electrical length to properly form the beams.

Similarly, the dummy and beam transmission line controls should be adjusted to produce reasonable lines that do not intersect and are smooth.

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Remcom RLD v1.7

Setting the transmission line geometry to the settings above will produce a lens that should look similar to that shown below.

43

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Remcom RLD v1.7

Next, the user is ready to evaluate the lens performance. To do this click Tools->Plot->Array Factor, and Tools->Plot->Beam Ports->Beam to Array Phase Errors. After re-sizing the widows for easy viewing, click the Calculate Beam Port Selector on the Phase Error plot to select All. This will display the Phase Error for each of the beam ports.

Click the Physical properties tab to have access to the Focal Ratio control. Adjusting this parameter while minimizing the error reported by Phase Error plot will produce a Focal ratio of approximately 1.0373. Coarse adjustments can be made with the slider but fine adjustment is best using the edit box.

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Remcom RLD v1.7

Note: RLD was designed to give the user maximum range of variability and control. Thus the user can create lens geometries which are not well formed or are physically impossible. Generally, when the input parameters go out of range or have reached some critical limit, RLD will warn the user of possible or fatal failures in the lens calculation. It is not a serious problem, only that one or more of the parameters is out of range. It is the combination of parameters which is in question not any one parameter. Below is an example of the warning message if the lens geometry is impossible to generate. Typically, choosing a Lens Width that is much smaller than the array length will result in such an error.

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Remcom RLD v1.7

Other types of warning messages are displayed in the Message box if, for example the geometry is self intersecting or Aperture distribution is unattainable with current settings.

Warning messages are always available but the message window can be toggled on or off by clicking View->Messages. Likewise another informational tool available is the port info which is toggled on and off using View->Port Info. This will enable and disable viewing the port data when moving the mouse cursor over a Rotman Lens port.

To determine what the Array Factor is for this set of parameters, Click the Electrical Properties Tab, then click the Beam Port Excitation Selector and select User Defined. Click Edit to open the excitation editor as shown.

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Remcom RLD v1.7

Click Zero All to clear the excitations. Then click the Magnitude for one of the beams, say 4, the central beam. This will highlight the “mag” entry for the 4th Beam port. Type 1 (1 volt) and click Ok. The Array Port plot should look like the following:

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Remcom RLD v1.7

To Scan the beam, click the Edit button for the Aperture Distribution once again, and select port 4 magnitude and reset to zero. Select port 1 and enter 1 volt and repeat for port 7. Click Ok. This will place two beams at +/- 20 degrees as specified. Below is the Array Factor after zooming in, showing the two +/- 20 degree beams.

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Remcom RLD v1.7

To determine the effects of Beam pointing watch the array factor and Aperture Coupling Magnitude plots while enabling/disabling the beam port pointing. Below are the resulting coupling amplitudes with and without Beam port pointing enabled respectively where the aperture distribution has been set to Manual from Uniform.

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Remcom RLD v1.7

Notice how the amplitude distribution along the Array contour is much more uniform with Beam port pointing enabled. Depending on the shape of the lens, this can be very useful to minimize spillover. Similar behavior can be seen for certain lens geometries using Array port pointing.

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Remcom RLD v1.7

The user may want to perform further tuning and optimization using the various parameters, focal length, element spacing, etc. Also, the Array amplitude and Phase distributions can be plotted, as described earlier.

In any of the plot types, the Legend on the left can be used to delete a particular data set from the plot, or de-activate it by clicking its check box:

By right clicking on a particular Legend entry, a menu will appear to Save, Remove, or Change the color of a particular trace on a plot as shown below.

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Saving a particular trace is useful for later comparison with other lens data by simply

importing the data using the file open button.

S-Parameters

RLD has an s-parameter export function. S-parameters are not updated continually like the other output data due to the computational overhead of computing all of the ports each time adjustments are made. The numbering scheme described in section 1 is important to keep in mind when analyzing s-parameters. The following diagram depicts how RLD treats the s-parameters for ports 4,20.

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Remcom RLD v1.7

To export the S Parameters for this lens, click File->Export-> S Parameters. This will open the s parameter dialog window:

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Browse or type the location of the location you would like to save the s-parameters, select any or all of the beam ports. Enter the min and max frequencies and the number of frequencies to compute the s-parameters.

The file or files selected will be created and can be used for further post-processing or viewing in RLD. To view them, click Tools->Plot->S-Parameters-> S-Parameter Magnitude. This will open the s-parameter plot window:

Clicking will open a browser where the user selects which file to plot:

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Select the file to plot and click open to view the data. It may be necessary to zoom in to see the data properly as described on page 21. Note the s-parameter plots are static and will not update as the other output parameters do. For each new lens, s-parameters must be exported and replotted.

At this point the user may wish to export the geometry in 2ds format to XFdtd. This accomplished using the procedure in section 5.

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Insertion Loss

The insertion loss of the lens will be calculated by summing the received powers atthe array and beam ports, relative to the transmitted power for each beam. The insertionloss calculation is calculated from the S parameters and is given by:

2k nk

n

L = -10 log |S | ∑

This is the insertion loss corresponding to beam port k where n is the index for the arrayports. The insertion loss is calculated for each beam as a function of frequency. Plots of insertion loss for each beam port and the S-parameters are available after post-processing due to the large amount of data for all of the possible port combinations preventing this from being done quickly in real time.

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7. Rotman Lens Designer – Theory and BackgroundRemcom Rotman Lens Designer RLD is a computer-aided design program for the design, synthesis, and analysis of Rotman Lenses and their variants. It is based on Geometrical Optics combined with the classical Rotman Lens design equations as in [1,2]. Since the intended usage for RLD is rapid synthesis of microwave lenses, various approximations and assumptions are made to allow for “real-time” design and tuning of the lens structure.

The analysis is based on geometry and geometrical optics and therefore assumes:

• Parasitic coupling is negligible.• Radiative leaks are not accounted for.• Transmission Line and material Dispersion is negligible• Dummy load effectiveness is assumed ideal

Effects which are accounted for include:

• Direct, line of sight propagation and Singly reflected rays between ports including sinc(u) radiation patterns for port tapers.

• Calculations of ray amplitude between ports include dielectric and conductor losses.• Individual port VSWR is approximated as that of the port to transmission line

transition.• Individual port losses are accounted for using analysis for the type of transmission

lines used.

Rotman Lens Contour Synthesis

The synthesis of the lens assumes several input parameters which are used to compute the inner contour (array contour) point as well as he line lengths. These are:

Element Spacing ( )η - The spacing of the linear array elements along the outer contour. The spacing is in free-space wavelengths or meters.

Focal Ratio (g) – The ratio of on axis focal length to off axis focal length [1] or G/F1. It is used to control the shape of the Focal arc.

Lens Width (G or F0) – The distance between the center of the Array contour and the center of the focal contour.

Scan Angle ( )α - This is the angle between the lens axis and one of the off-axis focal points and is the off axis beam angle for which phase error is zero [1].

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The lens inner contour points and transmission line lengths are solved for using the technique of path length comparison as in [1,2] .

1

2

0

2 2 21

2 2 22

2 2 20

( ) sin (0)

( ) sin (0)

( ) (0)

( ) ( cos ) ( sin )

( ) ( cos ) ( sin )

( ) ( ) ( )

F P W N N F W

F P W N N F W

F P W N G W

where

F P F X F Y

F P F X F Y

F P G X Y

α

α

α α

α α

+ + = +

+ − = +

+ = +

= + + −

= + + +

= + +

Lens Dimensions are then normalized by the off-axis focal length [1].

//

, , /( ) (0)

N FG G Fx y X Y F

W N WwF

η ==

=−=

Manipulation of these equations leads to the following relations for x,y,w,

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Remcom RLD v1.7

( )

2 2 2 2 20 0

2

22

0

2 2 202

0 0

2 2 4 420 0

20 0

(1 )2 2

0

11

1 12 2 2

4( )

y wx y a x w b waw bw c

where

gag a

g gb g b gg a g a

gb bcg a g a

ηη

η

η η

η η η

= −+ + = + −

+ + =

−= − − − − − = − + − − −

= − −− −

The lens design program solves for these points each time F, , ,gη α are modified.

Transmission Line Calculation

Once the lens contour is calculated and transmission line electrical lengths are determined, transmission lines for the chosen medium can be synthesized. The designer must choose the system impedance, transmission medium, Microstrip or Stripline, to determine the transmission line design parameters.

Transmission lines are then computed for each array port using the following steps:

• The dielectric parameters and thickness will be taken from the lens dielectric definition.

• The width of a line at the defined system impedance Z is calculated for the selected medium and is used to calculate the line effective dielectric constant. This is used to calculate the physical line length. This is calculated using standard microstrip/stripline design equations[3].

• The lens contour points determine the port spacing. • This width is used to calculate the correct impedance to match at the port end

of the transmission line.• A linear taper is used to connect the port to the line of system impedance

where the length is controlled by the acceptable VSWR.• A length of line is added to each taper to make it an integral number of

wavelengths at the center frequency.• The calculated length of the lens transmission lines is then added to the taper.

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The width of a system impedance transmission line is calculated for the defined medium and is used to calculate the line effective dielectric constant. This is used to calculate the physical line length.

For a microstrip lens the line width for the transmission lines is calculated using [3]:

2

8 22

12 0.611 ln(2 1) ln( 1) 0.39 22

A

A

r

r r

e Wfore dW

Wd B B B ford

επ ε ε

< −= − − − − + − + − >

where 0

0

1 1 0.110.2360 2 1

377

r r

r r

r

ZA

BZ

ε εε ε

πε

+ −= + + +

=

This ratio is used to calculate the effective dielectric constant:

1 1 12 2 1 12 /

r reff d W

ε εε + −= ++

Once the lens contour points have been calculated the port widths are known and the port impedance in the transmission line medium can be calculated using:

60 8ln( ) 14

120 11.393 0.667 ln 1.444

eff

eff

d W WforW d d

Z WfordW W

d d

ε

π

ε

+ ≤=

≥ + + +

For a stripline lens the line width for the transmission lines is calculated using [3]:

120 /

0.85 0.6 120 /r

r

x for ZWb x for Z

ε

ε

<= − − >

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where

30

r

xZ

πε

=

Once the lens contour points have been calculated the port widths are known and the port impedance in the transmission line medium can be calculated using:

300.441er

bZW b

πε

=+

where,

2

0 0.35

0.35 0.35

e

WforbW W

b b W Wforb b

>= − − <

For calculations of S-parameters and Array Factor, the loss of the transmission lines is also calculated. The loss of the connecting transmission lines is calculated as

Lc de whereα α α α− = +

and conductive attenuation coefficientdielectric attenuation coefficient

c

d

αα

==

where for microstrip lines [3],

0 r

/

k ( 1) tan /2 ( 1)

sc

o

ed

e r

R Np mZ W

Np m

α

ε ε δαε ε

=

−=−

and for stripline [3],

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30

0

00

2.7(10 ) 120 /30 ( )

/0.16 120 /

k tan /2

s rr

cs

r

d

R Z A Zb t

Np mR B Z

Z b

Np m

ε επ

αε

δα

− < − =

>

=

where,

2 1 21 ln ,

0.414 1 41 0.5 ln0.5 0.7 2

strip thickness, b = substrate thickness, and W = strip width.

W b t b tAb t b t t

b t WBW t W t

t

π

ππ

+ − = + + − −

= + + + +

=

Transmission line taper loss is approximated by calculating the loss of an equivalent length of transmission in the proper medium with impedance,

1 2Z Z Z=

using the equations 1.4-1.8.

Performance AnalysisPerformance Calculation

To calculate performance of the microwave lens, RLD approximates the coupling between ports using 2 dimensional aperture theory as is done in [8,24]. A sinc(u) approximation to the 2d beam pattern for each port is assumed which implies a uniform distribution to the port aperture. These port radiation patterns are used to compute the direct path and singly reflected path propagation from port to port as shown below.

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The port to port coupling is then approximated as,

( / 4)sin( )sin( )

sin2

j i j j kriij

i j

ii i

x w wxS ex x r

kwwhere x

π

λ

ϕ

− +=

=

g g

A similar calculation is done for Reflected paths but is done in two stages, the incident ray is calculated and then reflected according the properties of either the absorber boundary or dummy port boundary, and then the reflected ray path to the receiving port is calculated.

Phase Error is calculated as in [1,6] by comparing electrical lengths along two distinct paths from a given beam port through the lens. The first path travels through any one of the off-axis array ports, through its taper and transmission line, and finally along the path from the array element phase center (14% in from the end of the taper) to the beam phase front. The second path begins at the same beam port but travels through the center of the array curve and through a length of line common to all array ports. The comparison of these electrical lengths obtains the phase error for this beam port. This is done over the list of beam ports to produce a phase error plot as shown.

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0

=,

( )sin( )i

b a

b b TL i

a a TL

Phase Error R RwhereR R y n

R R

φ

φ α

φ

∆ −

= + +

= +

Aperture Coupling Magnitude and Phase Calculation

The aperture coupling is a plot of amplitude or phase distribution from a given beam port over the array ports. It is computed using the linear distance between the ports in the dielectric medium chosen. It can be used to optimize the beam pointing for best illumination and minimal spill-over as in section 6.

S-Parameter Calculation

The s-parameters are calculated by taking the complex ratio of the element voltages to the voltage at the excited port (one at a time). The assumption is that all other ports are perfectly

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terminated. The calculation of Sii (return loss) is simply converted from the estimated VSWR for the user-defined flare angle. All effects due to transmission line conductivity, length and impedance mismatch are accounted for. Transmission line dispersion is not accounted for as well as line discontinuities such as bends or mutual coupling.

Array Factor

The Array factor calculation is a textbook summation for a non-uniform linear array as in [34]. The complex element voltages are obtained similar to the s-parameters calculation and take into account all of the same effects.

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8. Rotman Lens Designer – References[1]. Rotman, W. and Turner, R., “Wide-Angle Microwave Lens for Line Source

Applications,” IEEE Transactions on Antennas and Propagation, vol. 11, no. 6, pp. 623-632, Nov. 1963.

[2]. Hansen, R.C., “Design Trades for Rotman Lenses,” IEEE Transactions on Antennas and Propagation, vol. 39, no. 4, pp. 464-472, April 1991.

[3]. Pozar, D.M., Microwave Engineering 2nd edition, New York: J. Wiley and Sons, 1998

[4]. Gagnon, David R., “Procedure for Correct Refocusing of the Rotman Lens According to Snell’s Law,” IEEE Transactions on Antennas and Propagation, vol. 37, no. 3, pp. 390-392, March 1989.

[5]. Musa, L. and Smith, M., “Microstrip Rotman Lens Port Design,” Antennas and Propagation Society International Symposium, vol. 24, pp. 899-902, June 1986.

[6]. Singhal, Pramod K., Gupta, R.D. and Sharma, P.C., “Recent Trends in Design and Analysis of Rotman-Type Lens for Multiple Beamforming,” Int. J. RF and Microwave Computer Aided Engineering, CAE8, pp. 321-338, 1998.

[7]. Simon, Peter S., “Analysis and Synthesis of Rotman Lenses,” American Institute of Aeronautics and Astronautics, 22nd AIAA International Communications Satellite Systems Conference & Exhibit 2004, pp. 1-11, May 2004.

[8]. Simon, Peter S., “Tools for Synthesis and Analysis of Rotman Lenses,” IEEE MTT-S Buenaventura Section Presentation, pp. 1-40, September 2003.

[9]. Kales, M. and Brown, R., “Design Considerations for Two-Dimensional Symmetric Bootlace Lenses,” IEEE Transactions on Antennas and Propagation, vol. 13, no. 4, pp. 521-528, July 1965.

[10]. Sletten, Carlyle J. Reflector and Lens Antennas : Analysis and Design Using Personal Computers. Norwood, MA: Artech House, Inc, 1988.

[11]. Claborn, K.D., Gallant, J.A., Willey, R.E. and Sinsky, A.I, “Computer-Aided Design of an Electrically Scanned Rotman Lens,” AP-S Session 12, 1040, pp. 353-356, June.

[12]. Rappaport, Carey M. and Zaghloul, Amir, I., “Optimized Three-Dimensional Lenses for Wide-Angle Scanning,” IEEE Transactions on Antennas and Propagation, vol. AP-33, no 11, pp. 1227-1236, November 1985.

[13]. Gent, H., “The bootlace aerial,” Royal Radar Establishment J., pp. 47-57, October 1957.

[14]. Rotman, W. and Turner, R.F., “Wide Angle Microwave Lens for Line Source Applications,” IEEE Trans. Antennas Propagation, vol. AP-11, pp. 623-632, November 1963.

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[15]. Ruze, J., “Wide Angle Metal Plate Optics,” Proc. IRE, vol. 38, pp. 53-59, January 1950.

[16]. Archer. D, “Lens-fed Multiple Beam Arrays,” Microwave J., pp. 37-42, October 1975.\

[17]. Rao, J.B.L., “Multifocal Three-Dimensional Bootlace Lenses,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 1050-1056, November 1982.

[18]. Kales, M.L, and Brown R.M, “Design Considerations for Two-Dimensional Symmetric Bootlace Lenses,” IEEE Trans. Antennas Propagat., vol. AP-13, pp. 521-528, July 1965.

[19]. Smith, M.S., “Design Considerations for Ruze and Rotman Lenses,” Radio Electron, Eng., vol. 52, no. 4, pp. 181-187, April 1982.

[20]. Gans, Michael J. and Amitay, Noach, “Narrow Multibeam Satellite Ground Station Antenna Employing a Linear Array with a Geosynchronous Arc Coverage of 60°, Part II: Antenna Design,” IEEE Transactions on Antennas and Propagation, vol. AP-31, no. 6, pp. 966-972, November 1983.

[21]. Shelton, J. Paul, “Focusing Characteristics of Symmetrically Configured Bootlace Lenses,” IEEE Transactions on Antennas and Propagation, vol. AP-26, no. 4, pp. 513-518, July 1978.

[22]. Singhal, P.K., Gupta, R.D., Sharma, P.C., “Theoretical Investigations on Elliptical Refocusing of Rotman-Type Lens for Multiple Beamforming,” Journal of Microwaves and Optoelectronics, vol. 3, no. 4, pp. 111-128, April 2004.

[23]. Katagi, Takashi, Mano, Seiji and Sato, Shin-Ichi, “An Improved Design Method of Rotman Lens Antennas, IEEE Transactions on Antennas and Propagation, vol. AP-32, no. 5, pp. 524-527, May 1984.

[24]. Maybell, Michael J., “Ray Structure Method for Coupling Coefficient Analysis of the Two Dimensional Rotman Lens,” Antennas and Propagation Society International Symposium, vol. 19, pp. 144-147, June 1981.

[25]. Wiebach, Wolfgang and Rausch, Ekkehart O., “Improving the Sidelobes of Arrays fed by Multiple-Beam Beam Formers,” Proceedings of the 1998 IEEE Radar Conference, pp. 313-318, May 1998.

[26]. Archer, Donald, “Lens-Fed Multiple-Beam Arrays,” Electronic Progress, vol. XVI, no. 4, Winter 1974.

[27]. Singhal, P.K., Sharma, P.C. and Gupta, R.D., “An Overview of Design and Analysis Techniques of Rotman Type Multiple Beam Forming Lens and Some Performance Results,” IE Journal-ET, vol. 84, pp. 52-58, January 2004.

[28]. Smith M.S. and Fong A.K.S., “Amplitude Performance of Ruze and Rotman Lenses,” The Radio and Electronic Engineer, vol. 53, pp. 329-336, September 1983.

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[29]. Peterson, Andrew F. and Rausch, Ekkehart O., “Scattering Matrix Integral Equation Analysis for the Design of a Waveguide Rotman Lens,” IEEE Transactions on Antennas and Propagation, vol. 47, no. 5, pp. 870-878, May 1999.

[30]. Cruz, J.L., Gimeno, B., Navarro, E.A. and Such V., “Diffraction by a Rotman Lens,” J. Optics (Paris), vol. 25, n. 3, pp. 115-120, 1994.

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